Please write your solutions on one side only of each
piece of paper you use, and please begin each solution on a new
sheet of paper. You may use factorial notation as well as
combination and permutation notation where appropriate (i.e.,
there is no need to expand 24!).

You are to work alone on this test. You may not use anyone
else's work nor may you refer to any materials as you complete the
test. You may ask me questions.

Evaluation Criteria

You may earn up to 10 points on each of questions 1 through 6.
For each question:

6 points count toward a correct solution to the problem. I
will evaluate the mathematics you use:

Is it accurate and appropriate?

Have you provided adequate justification?

4 points count toward how you express your solution. I will
evaluate how you communicate your results:

Is your solution clear and complete?

Have you expressed logical connections among components
of your solution?

1.

Respond to each of these questions. While you may show
steps leading to your solution, you do not need to generate
written explanations for questions (a) through (e) on this
page. (2 points each)

(a) What value K satisfies the equation P(12,4) =
KoC(12,4)?

(b) How many distinct arrangements exist for the letters
in the word mammilliform?

(c) In the expansion of (a+c+e+g+i)^10, state:

(i) the number of uncollected terms

(ii) the coefficient R in the collected term
Ra^3cg^2i^4.

(d) Determine the number of collected terms in the
expansion of (x+y)^6.

Geovanna walks from her apartment to the library every
day. Her apartment is 7 blocks north and 12 blocks east of
the library. Geovanna walks along city streets that are laid
out in a grid system.

(a) Given access to all the streets in the system, how
many different 19-block routes could Geovanna use to get
from home to the library? (3 points)

(b) The east/west street running outside Geovanna's
apartment is under repair. She cannot walk west on that
street for the first 3 blocks outside her apartment. Now how
many different 19-block routes could Geovanna use to get
from home to the library? (3 points)

(c) Again given access to all the streets in the system,
how many different 21-block routes could Geovanna use to get
from home to the library? (4 points)

Roger is taking a True/False test in his automobile
mechanics class. There are 10 unique items on the test.
Using a numbered answer sheet, students are requested to
circle either True or False corresponding to each item.

(a) How many different responses could Roger submit on
the 10-item answer sheet, assuming he responds either True
or False to each item? (3 points)

(b) If we allow for the possibility that Roger might not
circle either True or False on any or all items (i.e., he
could leave items blank), how many different answer-sheet
responses could he submit? (3 points)

(c) Roger's friend Howie took the True/False test before
Roger did. Howie told Roger there were 5 True items and 5
False items. If Roger heeded Howie's advice and circled 5
True and 5 False on the answer sheet, how many different
answer-sheet responses could he submit? (4 points)

Choose one of the following problems and solve it in the
space provided. If more than one solution appears, I will
evaluate only the first one I encounter.

I. A shelf is to contain nine different books, six
different paperback books and three different hardback
books. If the paperback books must be shelved in pairs (that
is, exactly two paperback books must be adjacent to each
other), how many ways can the nine books be shelved?

II. In the expansion of (r + s + t + u + v)^15, determine
the number of different ways a coefficient of 15 appears
among the collected terms.